Observe that the overall winner can be declared as soon as one party's LCB exceeds the UCB of all the other parties. On the other hand, when the winner is yet to be identified, one would ideally like to focus on “potential” winners, rather than query a constituency whose result does not seem relevant to
the big picture. Taking cue from the LUCB algorithm for bandits [1]}, the first step under DCB is to identify two contenders for the top position: party \(a^{t} = \operatornamewithlimits{argmax}_{i \in \lbrace 1, 2, \dots , K\rbrace } (\text{wins}^{t}_{i} + \text{leads}^{t}_{i})\) , and party \(b^{t} = \operatornamewithlimits{argmax}_{i \in \lbrace 1, 2, \dots , K\rbrace  \setminus \lbrace a^{t}\rbrace } \text{UCB}^{t}_{i}\) . Optimistic that sampling can reveal a win for \(a^{t}\)  and a loss for \(b^{t}\) —which would take us closer to termination—DCB picks one “promising” constituency each for \(a^{t}\)  and \(b^{t}\) . These constituencies, denoted \(c^{t}_{1}\)  and \(c^{t}_{2}\) , are defined below for use with both 1v1 and 1vr confidence bounds. The idea is the same: \(c^{t}_{1}\)  is the constituency in which \(a^{t}\)  appears poised to win by a large margin, and \(c^{t}_{2}\)  is the constituency in which \(b^{t}\)  appears poised to lose by a large margin.
\(c^{t}_{1} &={\left\lbrace \begin{array}{ll}\operatornamewithlimits{argmax}_{c} \min _{j \in \lbrace 1, 2, \dots , K\rbrace } (\text{UCB}(c, a^{t}, j) - \text{LCB}(c, a^{t}, j)) & \text{(1v1),} \\\operatornamewithlimits{argmax}_{c} \min _{j \in \lbrace 1, 2, \dots , K\rbrace } (\text{UCB}(c, a^{t}) - \text{LCB}(c, j)) & \text{(1vr),}\end{array}\right.}\\c^{t}_{2} &={\left\lbrace \begin{array}{ll}\operatornamewithlimits{argmax}_{c} \max _{j \in \lbrace 1, 2, \dots , K\rbrace } (\text{UCB}(c, j, b^{t}) - \text{LCB}(c, j, b^{t})) & \text{(1v1),} \\\operatornamewithlimits{argmax}_{c} \max _{j \in \lbrace 1, 2, \dots , K\rbrace } (\text{UCB}(c, j) - \text{LCB}(c, b^{t})) & \text{(1vr).}\end{array}\right.}\) 
